On commutative finite-dimensional nilalgebras
The authors survey certain results related to Albert's question: is every finite-dimensional commutative power-associative nilalgebra over a field solvable? In particular, the question has been answered in the affirmative for algebras of dimension \(\leq8\) provided that the characteristic of the ground field is different from 2, 3, and 5. In addition, the paper collects elementary proofs (mainly due to \textit{B. Mathes} et al. [Linear Algebra Appl. 149, 215--225 (1991; Zbl 0722.15016)]) for some results by \textit{M. Gerstenhaber} [Am. J. Math. 80, 614--622 (1958; Zbl 0085.26204)] and other authors about linear spaces of nilpotent matrices. These results are relevant in the context of Albert's question since for \(a\in A\), where \(A\) is a commutative power-associative nilalgebra of dimension \(n\) and nilindex \(t\), the left multiplication operator \(L_a: A\to A\) defined by \(L_a(x):=ax\) is known to be nilpotent of index \(\leq 2t-3\) [\textit{M. Gerstenhaber}, Duke Math. J. 27, 21--31 (1960; Zbl 0213.04101)]. Hence the linear space \(\{L_a\mid a\in A\}\) is isomorphic to a linear space of nilpotent \(n\times n\)-matrices.
scientific article; zbMATH DE number 6649138
scientific article